Dualities in Field Theories  

In Electromagnetic Theory > s.a. electromagnetism / differential forms; integrable systems; theory of physical theories [equivalence].
\(\def\se{_{\rm e}}\def\sm{_{\rm m}}\) * Hodge duality: The transformation \(F^{~}_{ab} \mapsto {}^*F^{~}_{ab}\), defined by \({}^*F^{~}_{ab} = {1\over2}\,\epsilon^{~}_{ab}{}^{cd}F^{~}_{cd}\), with \(\epsilon^{0123} = 1\).
* Continuous duality transformations: The 1-parameter family of transformations of the electric and magnetic quantities \(\{(\rho\se,\rho\sm),(j\se,j\sm),(E_i,H_i),(D_i,B_i)\}\); For a generic pair \((x\se,x\sm)\) of quantities, the duality is defined by

\[ \left(\matrix{x'\se \cr x'\sm}\right) = \left(\matrix{\cos\xi & \sin\xi \cr-\sin\xi & \cos\xi}\right) \left(\matrix{x\se \cr x\sm}\right), \quad {\rm or}\quad F'_{ab} = \cos\xi\, F^{~}_{ab} + \sin\xi\, {}^*F^{~}_{ab}\]

for the electromagnetic field, where ξ is a constant parameter (which cannot be promoted to a local one as in the gauge-theory trick).
* Properties: The values of \({\bf E} \times {\bf H}\), \({\bf E}\cdot{\bf D} + {\bf B} \cdot {\bf H}\), \(T_{ab}\), and the Maxwell equations, are left invariant.
* Applications: Duality between the Aharonov-Bohm and Aharonov-Casher effects.
@ General references: Misner & Wheeler AP(57); Montonen & Olive PLB(77) [monopole]; Zhu JPA(89) [complexions]; Anandan gq/95 [topological phases]; Deser & Sarioglu PLB(98)ht/97 [and Lorentz invariance]; Igarashi et al NPB(98)ht; Hatsuda et al NPB(99)ht [invariant lagrangians]; Li & Naón MPLA(01)ht/00; Przeszowski JPA(05)ht [light-front variables]; Julia ht/05-conf [generalization]; Barnich & Troessaert JMP(09)-a0812 [as bi-Hamiltonian system]; Bunster & Henneaux PRD(11)-a1011, Deser CQG(11)-a1012, Saa CQG(11)-a1101 [no local, gauged version]; Freidel & Pranzetti PRD(18)-a1806 [extension to the boundary]; Bunster et al PRD(20)-a1905 [and BMS invariance].
@ On a general manifold, and gravity: Witten SelMath(95)ht [on a general manifold]; in Garat JMP(15)gq/04; Bakas a0910-proc [and gravity]; Agulló et al PRD(14)-a1409, PRL(17)-a1607, PRD(18)-a1810 [anomaly, in curved spacetime].
@ In quantum electrodynamics: Buhmann & Scheel PRL(09)-a0806 [macroscopic]; Yang & Xu a2103 [no new conservation law].
@ In non-linear electrodynamics: Gibbons & Rasheed NPB(95)ht; Gaillard & Zumino LNP(98)ht/97, ht/97 [non-linear].
@ From Lagrangian: Bhattacharyya & Gangopadhyay MPLA(00)ht/98; Bliokh et al NJP(13) [helicity, spin, momentum and angular momentum].
@ Related topics: Fayyazuddin a1608 [3D theory with massive photon]; Castellani & De Haro a1803-in [fundamentality and emergence].

Gauge-Gravity Duality > s.a. AdS-cft correspondence; approaches to quantum gravity; Double Copy; holography.
* Idea: A set of relationships (for example the AdS-cft correspondence) which convert difficult problems in certain types of gauge theories into (relatively) simple geometric problems in gravity in one higher dimension.
@ General references: 't Hooft NPB(74) [precursor]; Polchinski a1010-ln; Shuvaev a1106; Maldacena a1106-ch; Billó et al a1304-proc [non-perturbative aspects]; Ammon & Erdmenger 15; De Haro SHPMP(15)-a1501 [and emergent gravity]; De Haro et al FP(16)-a1509 [rev]; Engelhardt & Fischetti CQG(16)-a1604 [boundary causality]; DeWolfe a1804-ln [intro]; Semenoff a1808-ln [holographic duality of gauge fields and strings].
@ And quantum gravity: Engelhardt & Horowitz IJMPD(16)-a1605-GRF; Hanada & Romatschke JHEP(19)-a1808 [simulations and phases].
@ And condensed-matter physics: Sachdev ARCMP(12)-a1108; > s.a. 2012 talk by Gary Horowitz on using the duality to model high-T superconductors.
@ Other applications: Wadia MPLA(10); Hossenfelder PRD(15)-a1412 [analog systems].

In Other Theories > s.a. hamiltonian systems; higher-order lagrangians; lagrangian dynamics; M-theory.
* In quantum mechanics: What Isidro calls duality is in reality an ambiguity in the choice of complex structure used in quantizing a classical theory.
@ General references: Savit RMP(80); Banerjee & Ghosh JPA(98) [chiral oscillator model]; Olive ht/02-proc; De Haro et al SHPMP-a1603 [and gauge symmetries]; McInnes NPB(16)-a1606 [field theories with no holographic dual]; De Haro & Butterfield a1707-in [schema, and bosonization example]; De Haro Syn(19)-a1801 [theoretical and heuristic roles]; Butterfield a1806-in [in physics vs philosophy]; Thompson a1904-proc [generalised dualities and their applications]; De Haro & Butterfield Syn-a1905 [and symmetries]; Turner PoS(19)-a1905 [in 2+1 dimensions]; De Haro a2004 [empirical equivalence].
@ Quantum mechanics: Isidro MPLA(03)qp, PLA(03)qp, qp/03-in [projective phase space], MPLA(04)qp/03 [torus phase space]; > s.a. coherent states.
@ Non-abelian gauge theory: Mohammedi ht/95; Duff IJMPA(96) and IJMPD(96) [in supersymmetric gauge theory, from strings]; Martín MPLA(99) [in path space]; Chan & Tsou IJMPA(99)ht; Tsou ht/00-ln, ht/00-conf; Faddeev & Niemi PLB(02)ht/01 [SU(2) Yang-Mills theory]; Majumdar & Sharatchandra IJMPA(02); Deser & Seminara PLB(05)ht [duality invariance for free bosonic and fermionic gauge fields]; Kihara JMP(11) [generalized self-duality equations]; Ho & Ma NPB(16)-a1507.
@ Sigma-models: Mohammedi et al ZPC(97)ht/95; Mohammedi PLB(96)ht/95, PLB(96).
@ Linearized gravity: Henneaux & Teitelboim PRD(05)gq; Barnich & Troessaert JMP(09)-a0812 [as bi-Hamiltonian system]; Bakas a0910-proc; Troessaert a1312-PhD.
@ General relativity: Hawking & Ross PRD(95)ht [electric and magnetic black holes]; Maartens & Bassett CQG(98)gq/97; Nouri-Zonoz et al CQG(99)gq/98 [NUT]; Dadhich MPLA(99)gq/98, MPLA(99), GRG(00)gq/99; Abramo et al MPLA(03) [with scalar field]; Deser & Seminara PRD(05)ht [failure in non-linear case]; Julia ht/05-conf; da Rocha & Rodrigues JPA(10)-a0910 [in gravitational theories]; Dehouck NPPS(11)-a1101 [and supergravity].
@ Scale factor duality in cosmology: Clancy et al CQG(98)gq; Di Pietro gq/01/MPLA [quintessence]; Harlow & Susskind a1012 [general criteria].
@ In string theory: Rickles SHPMP(11); Polchinski SHPMP(17)-a1412; Huggett & Wüthrich a2005-ch [meaning and significance]; > s.a. strings.
@ Related topics: Gaona & García IJMPA(07) [first-order actions]; Lindström et al JHEP(08)-a0707 [T-duality for generalized Kähler geometries]; Barnich & Troessaert JHEP(09)-a0812 [for spin-2 fields in Minkowski space]; Nussinov et al NPB(15)-a1311 [dualities as conformal transformations, and practical consequences]; Miyaji & Takayanagi PTEP(15)-a1503 [codimension-two spacelike surfaces and states in dual Hilbert spaces]; Sourlas a1907 [fermionic theories].
> Related topics: see gravitomagnetism; self-dual fields [connections] and self-dual solutions in general relativity [Weyl tensor].
> Other dualities: see Galerkin Duality.

Dual Mass in Gravitation
@ General references: Lubkin IJTP(77); Magnon JMP(87), NCA(88); Torre CQG(95)gq/94.
@ Phenomenology: Cates et al GRG(88); Rahvar & Habibi ApJ(04)ap/03 [microlensing signatures]; Danehkar HEPGC(17)-a0707.

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